Geodesic
Quantum Mapping Class Group, Pentagon Relation, and Geodesics
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Mathematical physics. Problems of quantum field theory, Tome 226 (1999), pp. 163-179

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The mapping class group transformations that satisfy the pentagon relation are constructed for classical and quantum Teichmüller spaces coordinatized in terms of graphs. Classical and quantum geodesic algebras governed by the skein relations are discussed. 
@article{TM_1999_226_a12,
     author = {V. V. Fock and L. O. Chekhov},
     title = {Quantum {Mapping} {Class} {Group,} {Pentagon} {Relation,} and {Geodesics}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {163--179},
     publisher = {mathdoc},
     volume = {226},
     year = {1999},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_1999_226_a12/}
}
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V. V. Fock; L. O. Chekhov. Quantum Mapping Class Group, Pentagon Relation, and Geodesics. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Mathematical physics. Problems of quantum field theory, Tome 226 (1999), pp. 163-179. http://geodesic.mathdoc.fr/item/TM_1999_226_a12/